Delft University of Technology
New Approach to Multisite Downscaling of Precipitation by Identifying Different Set of
Atmospheric Predictor Variables
Basu, Bidroha; Nogal, Maria; O'Connor, Alan DOI
10.1061/(ASCE)HE.1943-5584.0001900 Publication date
2020
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Journal of Hydrologic Engineering
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Basu, B., Nogal, M., & O'Connor, A. (2020). New Approach to Multisite Downscaling of Precipitation by Identifying Different Set of Atmospheric Predictor Variables. Journal of Hydrologic Engineering, 25(5), [04020013]. https://doi.org/10.1061/(ASCE)HE.1943-5584.0001900
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New Approach to Multisite Downscaling of
Precipitation by Identifying Different Set
of Atmospheric Predictor Variables
Bidroha Basu
1; Maria Nogal
2; and Alan O
’Connor
3Abstract: Estimating reliable projections of precipitation considering climate change scenarios is important for hydrological studies. General circulation models provide future climate simulations at large scale in terms of large-scale atmospheric variables (LSAVs). Those LSAVs can be downscaled to finer special resolution using several downscaling approaches. This paper presents a support vector regression (SVR)-based downscaling approach to downscale rainfall at several locations in a study area. Because the rainfall generation mechanisms cannot be the same for all the sites in a study area, conventional multisite downscaling approaches that assume the same rainfall generation mechanism should not be used. Therefore, a new downscaling approach is proposed that (1) divides the study area in several climatological regions, and (2) develops different downscaling models for each of the climatological regions to obtain future projections of rainfall. The new approach was implemented on rainfall data obtained for Republic of Ireland to demonstrate the effectiveness of the approach compared with existing approaches. Future projections of rainfall were obtained for the period 2012–2050 corresponding to four Representative Concentration Pathway climate change scenarios. The performance of the SVR approach was compared with that of relevance vector machine– and deep learning–based downscaling approaches. DOI:10.1061/(ASCE)HE.1943-5584.0001900. © 2020 American Society of Civil Engineers.
Author keywords: Statistical downscaling; Rainfall; Support vector regression; Global K-means; Future rainfall projections.
Introduction
Precipitation is one of the most important hydrometeorological variables for hydrological modeling. Prediction of rainfall on a catchment scale due to the effect of climate change is a challenging problem. Predictors available at the large scale from general circu-lation models (GCMs) fail to capture the local- or catchment-scale variability present in the rainfall. To overcome this limitation, large-scale GCM outputs often are downlarge-scaled to finer spatial resolution. The downscaling approaches can be classified broadly into two categories, dynamic downscaling and statistical downscaling.
Dynamic downscaling considers the initial conditions, time-dependent lateral meteorological conditions, and surface boundary conditions along with the GCM data and transfers the large-scale GCM data to a higher-resolution regional scale by using regional climate models (RCMs). The advantage of those models is that they can account for the physical processes governing the rainfall and can simulate finer-scale atmospheric processes (e.g., orographic
precipitation) better than the host GCMs (Fowler and Wilby 2010). The disadvantages of RCMs are that the models are complex and require significant time to produce outputs (Eden and Widmann 2014). In addition, RCMs provide future climatic conditions at spe-cific grids, and hence it is not possible to directly obtain future rain-fall simulations in a region located slightly outside the RCM grids (D’Onofrio et al. 2014). In statistical downscaling, the regional climate at the local scale is considered to be a function of the large-scale climate denoted by the large-large-scale atmospheric variables. To perform the downscaling, a statistical relationship is developed between those large-scale atmospheric variables (e.g., temperature at high elevations, atmospheric pressure, wind speed, humidity, and solar radiation) and the predictand variable (e.g., rainfall or temper-ature at the watershed scale) at the local scale. Statistical downscal-ing has gained wide popularity due to its low computational cost and simplicity compared with dynamic downscaling (Okkan and Fistikoglu 2014;Rashid et al. 2016;Sachindra et al. 2016). Wood et al. (2004) considered linear interpolation, spatial disaggregation, and bias correction and spatial disaggregation (BCSD)-based stat-istical downscaling methods to simulate meteorological variables from 1975 to 1995 and estimated land surface energy and water fluxes using a variable infiltration capacity model. They noted that the performance of BCSD was superior to that of the other two statistical downscaling methods, and the dynamic downscaling did not provide any improvement compared with statistical downscal-ing. Ahmed et al. (2013) also used a BCSD-based statistical down-scaling approach to downscale precipitation and maximum and minimum temperature at the daily scale using six GCMs and com-pared it with four RCMs, and found similar results for both types of downscaling approaches. Application of multiple regression– based statistical downscaling and RCM dynamic downscaling of monthly precipitation at 42 stations in Sweden by Hellström et al. (2001) yielded similar performance in reproducing the seasonal pre-cipitation cycle. Landman et al. (2009) noted that a model output
1Research Fellow, School of Architecture, Planning and Environmental
Policy, Univ. College Dublin, Dublin D14 E099, Ireland; Adjunct Assistant Professor, Dept. of Civil, Structural and Environmental Engineering, Trinity College Dublin, Dublin D02, Ireland (corresponding author). ORCID: https://orcid.org/0000-0002-8822-7167. Email: bidroha.basu@ ucd.ie; bbasu@tcd.ie
2Assistant Professor, Materials, Mechanics, Management, and Design,
Delft Univ. of Technology, Delft 2628 CN, Netherlands. Email: m.nogal@ tudelft.nl
3Professor, Dept. of Civil, Structural, and Environmental Engineering,
Trinity College Dublin, Dublin D02, Ireland. Email: oconnoaj@tcd.ie Note. This manuscript was submitted on December 8, 2018; approved on October 18, 2019; published online on February 27, 2020. Discussion period open until July 27, 2020; separate discussions must be submitted for individual papers. This paper is part of the Journal of Hydrologic Engi-neering, © ASCE, ISSN 1084-0699.
statistics-based statistical downscaling method was slightly better at simulating the rainfall at 963 stations in South Africa compared with the RCM dynamic downscaling method. Various other studies (Murphy 1999;Wilby et al. 2000;Diez et al. 2005) found that stat-istical and dynamic downscaling methods produce similar outcomes in simulating the present and future climate.
The statistical downscaling can be classified in three categories, weather generator, weather typing, and transfer function–based methods (Wilby et al. 1998;Nguyen et al. 2006;Fowler et al. 2007;
Vrac and Naveau 2007). A weather generator (Wilks and Wilby 1999; Olsson et al. 2009) stochastically generates the future weather scenarios in terms of local-scale time series by using a stat-istical model (probability density functions) whose parameters are related to the large-scale data (e.g.,Vrac and Naveau 2007), whereas weather typing generates synthetic sequences of weather patterns from past observations by conditioning the simulation of small-scale data on weather types over the region of interest (e.g.,Vrac et al. 2007b). Neither of those two statistical downscal-ing methods is suitable for long-term rainfall projections. The third option is the transfer function–based downscaling approach, which translates the large-scale atmospheric information directly to the local-scale meteorological data using a statistical transfer function; it has been used by several researchers (Vrac et al. 2007a;Dibike et al. 2008;Olsson et al. 2004;Srinivas et al. 2014).
Multisite downscaling of rainfall is required for analyzing watersheds located in a complex terrain in which topography becomes a major factor in the generation of the rainfall process. Several studies in literature focused on multisite downscaling of rainfall (e.g.,Fowler et al. 2005;Haylock et al. 2006; Vrac and Naveau 2007;Wetterhall et al. 2006). However, those studies as-sumed that the rainfall generation process was the same for the entire study area. Those studies considered the same set of large-scale atmospheric variables to develop a unique downscaling model for the entire study area to downscale precipitation. In a real-world scenario, the rainfall generation process is expected to change when the study area is considerably large in size. Few available approaches can address this issue. This study developed an alter-native approach to identify different sets of large-scale atmospheric variables (LSAVs) at different regions of the study area and to use those LSAVs to develop different statistical downscaling models for each of those regions to obtain future projections of rainfall for the sites located in each region. The study used a global K-means (GKM) clustering algorithm to form clusters (climatological re-gions) in the study area and subsequently used a transfer function [support vector regression (SVR)]-based statistical downscaling ap-proach to develop different downscaling models for each of the identified climatological regions. The performance of the proposed alternative approach was compared with that of the existing ap-proaches which develop a unique downscaling model by assuming the same set of large-scale atmospheric variables for the entire study area. The advantages of the proposed method in downscaling rainfall were demonstrated by application to 464 rain gauges in the Republic of Ireland.
Description of Study Area and Data
The study considered 464 rain gauge stations having daily rainfall obtained from MET Eireann (Éireann 2009). The rain gauges were selected by ensuring that each gauge had at least 5 years of records between January 1979 and May 2016. The locations of the rain gauges are shown in Fig.1along with the elevation of the Republic of Ireland.
The goal of the study was to obtain future projections of rainfall considering the effect of climate change. For this purpose,
European Centre for Medium-Range Weather Forecasts ReAnaly-sis (ERA) Interim data, which are available from January 1979, were used.
A set of LSAVs that can affect the rainfall events was down-scaled at a 2° × 2° grid scale covering the Republic of Ireland. The grids ranged from 51°N to 57°N and from 5°W to 11°W, and consisted of nine grids covering the Republic of Ireland. The three-dimensional LSAVs are available at different pressure levels. For this study, LSAV values at 17 pressure levels [100 (1,000), 92.5 (925), 85 (850), 70 (700), 60 (600), 50 (500), 40 (400), 30 (300), 25 (250), 20 (200), 15 (150), 10 (100), 7 (70), 5 (50), 3 (30), 2 (20), and 1 (10) kPa (mb)] were considered for analysis.
Proposed Methodology
This section presents the support vector regression–based rainfall downscaling approach. Subsequently, details of the GKM cluster-ing used to identify climatological regions are provided. Followcluster-ing this, the theoretical background of the SVR approach and the pro-cedure for obtaining future projections of rainfall is explained.
SVR Approach to Multisite Downscaling of Rainfall The proposed methodology involves the following steps (Fig.2): 1. The predictor variables are identified from large-scale atmo-spheric variables available in the observed/reanalysis data and GCM simulations, such that they are reasonably well correlated with historically observed rainfall (predictand) at all the target sites in the study area. In situations in which the study area is large, the sites in the study area should be delineated in clusters/ climatological regions. The correlation vector between the his-torically observed rainfall data and the identified LSAVs corre-sponding to each site, along with the location indicators (latitude and longitude and elevation of the rain gauges), can be used as attributes to form climatological regions using a clustering technique. This study used partition-based global K-means clus-tering (Likas et al. 2003) to form the climatological regions, be-cause the climatological regions must exhibit hard boundaries. Advantages of GKM are that, unlike conventional K-means clustering, the GKM is not sensitive to the initial conditions and it can identify the global optimal solution while optimizing the objective function to identify clusters. Other hard clustering techniques such as entropy-based clustering or hierarchical clustering can be an alternative option to form climatological regions. Most of the clustering techniques fail to form efficient clusters when the data used to form clusters are nonlinear and the regions/clusters cannot be separated by using linear planes. The objective in forming climatological regions is to in-crease the intersite correlation between the predictands in each region. In situations in which observations of LSAVs are una-vailable, reanalysis data can be used as a surrogate for the analysis. To obtain future projections of rainfall (predictand), general circulation model simulations can be used, which pro-vide the simulated values of LSAVs for the future in different climate change scenarios. The GCM data are available at various time scales, such as 6-hourly, daily, and monthly scales. How-ever, GCM simulations available at finer time scales are not considered reliable (Prudhomme et al. 2002, p. 1,138;Brown et al. 2008, p. 20), and simulations at coarser time scales (e.g., monthly) are preferred. If the spatial resolution of reana-lysis data and GCM data are different, the GCM data should be spatially interpolated to the resolution of the reanalysis data using software such as GrADS (Doty and Kinter 1993).
2. Once the clusters are formed, sets of predictor variables for each cluster are identified using the correlation between the LSAVs and the rainfall data. The identified predictor variables henceforth are referred to as large-scale atmospheric predictor variables (LSAPVs). The LSAPVs for separate climatological regions can be different from each other.
3. To develop the SVR-based downscaling model, the available historical data are divided into two subgroups. The first sub-group is called the calibration set, and the second is called the validation set. The SVR model is developed using data from the calibration set, and the performance of the model is tested based on data from the validation set.
4. Each of the identified monthly LSAPV data sets from the cali-bration set corresponding to each of the regions is standardized by subtracting the respective mean and dividing by the respec-tive standard deviation. Standardization of the LSAPV is neces-sary to nullify the effect of differences in magnitude, range, and variance of values corresponding to the LSAPVs. Reduction of dimensionality of LSAPVs and intercorrelation between the LSAPVs can be achieved by preforming principal component analysis. However, because some information will be lost due to omission of some of the principal components, it was con-sidered in this study.
5. Development of the SVR relationship is established for every site in the region between the LSAPVs and the observed monthly rainfall data. Because the optimal parameters of the SVR model are not known a priori, a grid search procedure (Gestel et al. 2004) is used to identify the optimal values of those parameters for the cluster. The parameters for which the model output (downscaled monthly values of rainfall for all sites in the region) is the closest [quantified in terms of RMS error (RMSE) and Nash–Sutcliffe error (NSE)] to the contemporaneous ob-served monthly rainfall at the corresponding target sites are con-sidered to be the optimal. The selected optimal parameters are the same for all the sites in a region, and the developed SVR model is called the regional SVR downscaling model. The equa-tions to estimate RMSE and NSE are as follows:
RMSE¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xn i¼1 ð^yi− yiÞ2 s ð1Þ NSE¼ 1 − Pn i¼1ð^yi− yiÞ2 Pn
i¼1ðyi− ¯yiÞ2
ð2Þ where yii ¼ 1; : : : ; n) = observed monthly mean rainfall; n = number of observations; ¯yi = mean of yi observations; and
Fig. 1. Location of 464 rain gauges in the Republic of Ireland.
^yi= predicted monthly mean rainfall based on the downscaling model.
6. The developed regional SVR downscaling model then is vali-dated by using data from the validation set. For this purpose, each of the LSAPVs from the validation set is standardized by subtracting the respective mean and dividing by the respec-tive standard deviation. The standardized LSAPVs then are provided to the regional SVR downscaling model to obtain downscaled values of monthly rainfall outputs. The modeled outputs of monthly rainfall for each site in a region are com-pared with the corresponding values of observed rainfall for the validation period, and the performance of the modeled out-put is quantified in terms of the two performance measures, RMSE and NSE. In situations in which the model outputs are close to the observed rainfall, the RMSE should be small and the NSE should be close to unity. When the model-predicted rainfall for all sites in the study area is close to the observed rainfall values, the developed model can be used to obtain future projections of rainfall for those rain gauge sites. In situations in which the performance is poor, it can be concluded that the se-lected LSAPVs cannot model the rainfall process and a new set
of LSAPVs need to be identified for developing an effective downscaling model.
7. Once the model has been developed and validated, it can be used to obtain future projections of rainfall for all the sites in a cluster/climatological region by using the developed downscal-ing model for that region. The steps to obtain future projections of mean monthly rainfall are as follows:
a. The selected LSAPVs extracted from the GCM simulations for the historical period are collated. Because the values in GCM have inherent biases, it is necessary to perform a bias correction before the future projections of LSAPVs from the GCM simulations can be used to obtain future projections of rainfall. To perform bias correction, the equidistant quantile matching (EQM) bias correction approach (Li et al. 2010) is used. EQM-based bias correction explicitly considers the changes in the distribution of the future climate, including the upper tail of the distribution involving extreme rainfall events (Li et al. 2010). Furthermore, EQM bias correction is applicable to smaller as well as larger spatial domain, whereas most of the existing bias correction approaches were developed based on statistical moments or regression equa-tions and are constrained to local-scale application. b. The bias-corrected future projections of GCM-simulated
LSAPVs then are standardized and considered as an input to the developed SVR-based downscaling model to obtain future projections of monthly mean rainfall for each site in the cluster/climatological region.
Global K -Means Clustering Algorithm
Consider that there are N rain gauge stations in the study area. Furthermore, assume that the number of LSAVs considered is equal to l. Corresponding to every pressure level for every LSAV, the average correlation between rainfall and nine LSAV grids sur-rounding the study area are estimated. Those correlation values along with the three location details (latitude, longitude, and eleva-tion) are considered to create feature vector for each rain gauge station. Denote the feature vector as fzs0¼ ½zs;1; : : : ; z0 s;L0 ; s ¼ 1; : : : ; Ng, where L is the dimension of the feature vector. The fea-ture vectors are rescaled tofzs¼ ½zs;1; : : : ; zs;L; s ¼ 1; : : : ; Ng as zs;j¼ ðzs;j0 − ¯zjÞ=σj0 for1 ≤ j ≤ L; 1 ≤ s ≤ N ð3Þ where ¯zj0¼PN s¼1zs;j=N and σj0 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PN s¼1ðz0 s;j− ¯zjÞ=ðN − 1Þ0 q . Rescaling of the attributes is necessary to nullify influences asso-ciated with the magnitude and variance of their values in cluster formation.
The rescaled feature vectors Z ¼ ½z1; z2; : : : ; zNTare partitioned into Kopt clusters (climatological regions) using global K-means clustering algorithm. Details of the GKM steps that were adapted from Basu and Srinivas (2016) are provided in the Appendix.
Support Vector Regression
This section describes the procedure to develop the SVR relation-ship (Vapnik 1995) in terms of input vector (predictor) xt¼ ½xt;1; :::;xt;m ∈ ℜm, where m denotes the number of predictors, and output (predictand) yt∈ ℜ, corresponding to time t ¼ f1;2; :::;ng.
The relationship between xt and yt can be expressed as yt¼ fðxtÞ þ εt ð4Þ whereffð·Þ; Rm→ Rg = nonlinear transformation function; and εt = white noise whose expected value E½εt is zero.
Obtain Reanalysis grid climate data surrounding the study area and identify set of LSAVs that influence rainfall in the study area based on high correlation values between rainfall and LSAVs averaged over the selected grids
Use the average correlation obtained at different pressure level to form climatological regions based on GKM clustering procedure
Identify set of LSAPVs corresponding to identified pressure levels for each of the climatological regions based on the correlation between rainfall and LSAVs
Divide the LSAPV and rainfall data in the climatological region into calibration and validation sets and use the calibration set data to develop SVR based downscaling model after rescaling the LSAPVs
Use validation set data to validate the downscaled model. Develop different SVR based downscaling models for each climatological regions
Obtain LSAPVs for each climatological regions for the same grids and at same pressure level as that for the Reanalysis data. Perform bias correction of the GCM based LSAPVs using EQM approach
Use bias corrected GCM LSAPVs to predict future projections of rainfall using developed SVR based downscaling model
Fig. 2. Flowchart of the proposed methodology for multisite downscal-ing of rainfall.
Let the functionϕð·Þ map xt at a higher p-dimensional space, where a linear relationship exists betweenϕðxtÞ and yt. The linear relationship can be expressed as
½yt1×1¼ ½ϕðxtÞ1×p½wp×1þ ½b1×1 ð5Þ where weights
½wp×1¼ ½ w1 w2 · · · wpT ð6Þ The parameters w1; w2; : : : ; wp, and b can be estimated by op-timizing the following objective function:
Lðw; eÞ ¼ minw;b;e 1 2wTwþ C 1 2 Xn t¼1 e2t ð7Þ where error et¼ yt− ϕðxtÞw − b ð8Þ ½en×1¼ ½ e1 e2 · · · enT; and C = SVR parameter consid-ered to be the weight associated itho the sum of squares of error etat a given time. Choosing high values of the parameter reduces (increases) the error in model prediction by overfitting (underfit-ting) the model.
Minimization of the termð1=2ÞwTw ensures that the model is not overfitted, whereas minimization of the term Cð1=2ÞPn
t¼1e2t ensures that the model prediction error is not significantly high.
Introducing Lagrange multipliers, the optimization problem becomes
Lðw; b; e; αÞ ¼ Lðw; eÞ −Xn t¼1
αt½ϕðxtÞw þ b þ et− yt ð9Þ whereαt= Lagrange multipliers; and½αn×1¼ ½α1 α2 · · · αnT. The optimal solution can be obtained by optimizing the function Lðw; b; e; αÞ using the following conditions:
∂Lðw; b; e; αÞ ∂w ¼ 0; ⇒ w ¼ Xn t¼1 αtϕðxtÞ ð10Þ ∂Lðw; b; e; αÞ ∂b ¼ 0; ⇒ Xn t¼1 αt¼ 0 ð11Þ ∂Lðw; b; e; αÞ ∂et ¼ 0; ⇒ αt¼ C × et ∀ t ð12Þ ∂Lðw; b; e; αÞ ∂αt ¼ 0; ⇒ wTϕðxtÞ þ b þ et− yt¼ 0 ∀ t ð13Þ Based on the conditions in Eqs. (10)–(13), and eliminating w and e Xn t¼1 αt¼ 0 ð14Þ Xn t0¼1 αt0ϕðxt0ÞT·ϕðxtÞ þ b þαt C¼ yt; for ∀ t ¼ 1; : : : ; n ð15Þ where ϕðxtÞT·ϕðxtÞ = dot product of input vector in high-dimensional transformed space. The dot product can be expressed by a kernel function (Vapnik 1995) given as
ϕðxiÞT·ϕðxjÞ ¼ Kðxi; xjÞ ð16Þ This study considered a Gaussian radial basis kernel (RBF) function for analysis, which can be expressed as
Kðxi; xjÞ ¼ e−γkxi−xjk2; γ > 0 ð17Þ The RBF function was chosen instead of a polynomial because the RBF function gives access to any analytic functions, and it is parsimonious. For chosen values ofγ and C, values of fαt; t ¼ 1; : : : ; ng and b are estimated by solving Eqs. (9)–(12). Subsequently, future projections of the predictand ytf can be obtained as ytf¼ Xn t¼1 αtϕðxtÞT·ϕðxt fÞ þ b ¼ Xn t¼1 αtKðxt; xtfÞ þ b ð18Þ where xtf = future projection of predictor vector corresponding to time tf. From Eq. (12) it can be noted thatαt=c ¼ et, where et is the error, the expected value of which is assumed to be zero.
Results and Discussion
To identify the proper sets of LSAVs affecting rainfall for all the rain gauge stations in the study area, the correlations of rainfall data with each of the selected LSAVs at 17 pressure levels were esti-mated. In this study, 10 LSAVs were considered (Table 1). The average correlation between the rainfall and the surrounding 9 LSAV grids corresponding to each of the 10 variables and pressure levels are plotted in Figs.3(a–j).
A set of variables that were highly correlated with the rainfall data for the 464 rain gauges at their corresponding pressure levels were identified (Table1). Those identified LSAVs henceforth are called large-scale atmospheric predictor variables (LSAPVs).
One drawback of this approach to identify LSAPVs is that the same set of attributes was considered for entire study area (Repub-lic of Ireland). In a real-world scenario, it is unlikely that the rainfall is influenced by the same climatological processes at every location in the entire country. To address this issue, the correlations between the LSAVs with the rainfall data along with three location indica-tors (latitude, longitude, and altitude) were considered as attributes to form climatological regions in the Republic of Ireland. The global K-means clustering technique (Basu and Srinivas 2016;Bharath et al. 2016) was used to identify those regions, and the analysis showed that the study area can be divided into six clusters/ climatological regions (Fig.4). The first cluster is in the windward
Table 1. List of LSAPVs considered for 464 rain gauges No. V
and Fð·Þ LSAV −ve +ve
1 Specific humidity, q 200 100 2 Relative humidity, r 250 600 3 Temperature, t 700 200 4 U component of wind, u — 150 5 V component of wind, v — 925 6 Vertical velocity, w 500 — 7 Geopotential, z 1,000 —
8 Total column water, tcw — —
9 Mean sea level pressure, msl 1,000 —
10 Instantaneous surface
sensible heat flux, ishf
— —
Note: –ve (+ve) denote rainfall negatively (positively) correlated to the
LSAPV.
direction of the Wicklow Mountains located in the eastern part of the country, where the wind predominantly arrives from the southwest direction. The second cluster is located in the western part of the country near the Atlantic Ocean, at medium to high altitude (Fig.1). Cluster 3 is located in the central part of the country at low to medium altitude, Cluster 4 is located in the southern part at low
altitude, and Cluster 5 is located in the northern part of the country, where the altitude is relatively high. Cluster 6 ranges along the southeastern coastline of the country at low altitude. The number of stations and average record length in each of the clusters were as follows: Cluster 1—82 stations, 22.7 years; Cluster 2—97 sta-tions, 28.5 years; Cluster 3—98 stations, 26.1 years; Cluster 4—33
Fig. 3. Correlation plots between rainfall data for each of the 464 sites and LSAVs: (a) specific humidity (q); (b) relative humidity (r); (c) temperature (t); (d) U component of wind (u); (e) V component of wind (v); (f) vertical velocity (w); (g) geopotential (z); (h) total column water (tcw); (i) mean sea level pressure (msl); and (j) instantaneous surface sensible heat flux (ishf) corresponding to different pressure levels.
stations, 12.2 years; Cluster 5—63 stations, 27.7 years; and Cluster 6—91 stations, 30 years. Except for Cluster 4, which had the least number of stations, the average record of each clima-tological region had precipitation records covering more than 22 years.
Intersite cross correlation for the calibration and validation data set for 464 stations was prepared (Fig. S1). Comparison of the correlations indicated that the correlations between the rain gauge stations changed considerably from the calibration period
to the validation period due to the effect of climate change. The figures also indicate that the correlation structure between rain gauge stations in the study area varied considerably, justifying the requirement to develop different downscaling models in the study area. To demonstrate the advantage of forming climatological re-gions, correlation between sites in each of the six regions were plot-ted (Fig.S2). The figures in the calibration set indicate that most of the stations in those climatological regions had nearby correlations among themselves. The differences in intersite correlation between
Fig. 3. (Continued)
rain gauge stations for the validation period was found to be higher, which might be due to the effect of climate change or to other natu-ral or anthropogenic factors.
The large-scale atmospheric variables for each of those clusters then were identified by investigating the average correlations be-tween the rainfall and 9 grids for each of the 17 pressure levels. The identified LSAPVs for each of the clusters are provided in Table 2. The LSAPVs corresponding to each of the clusters (climatological regions) were different from each other.
Based on the selected LSAPVs, support vector regression– based downscaling models were developed for each of the clusters in the study area. Because the model parametersγ and C were not known a priori, a grid search procedure was used to obtain the optimal parameters for the study. For this purpose, the first 75% of the data set corresponding to each rain gauge for the historical period was used for calibration, and the remaining 25% of the data set was used for validation of the model. The number of data points (in months) for the calibration and validation sets for each of the
Fig. 3. (Continued)
464 stations was plotted as boxplot (Fig.S3). The majority of the stations had more than 150 months of data points in the calibration set and more than 50 data points in the validation set. Furthermore, the average records for the calibration and validation sets for stations corresponding to each of the climatological regions were 204.6 and 67.7 months (Cluster 1), 256.9 and 85.2 months (Cluster 2), 235.1 and 77.8 months (Cluster 3), 109.7 and 36.1 months (Cluster 4), 249.3 and 82.7 months (Cluster 5), and 270 and 90 months (Cluster 6), respectively. This indicates that,
except for Cluster 4, which consisted of only 33 stations, the ma-jority of stations had adequate number of records for developing the SVR-based downscaling model. Both SVR parameters were varied from 10 to 1,000 with an interval of 10 units, and the difference between the observed and model-predicted rainfall was measured in terms of two performance measures (NSE and RMSE). The op-timal parameters of the SVR model were selected when the model yielded reasonable performance, i.e., NSE closer to unity and RMSE close to zero. All the rain gauges in a region are not
Fig. 3. (Continued)
expected to have the lowest error corresponding to a given param-eter combination of SVR model. For this purpose, the paramparam-eter combination that was selected by the majority of the rain gauges in a climatological region was considered to be the optimal param-eters for a region. The values of the optimal paramparam-eters are pro-vided in Table3, and the developed model is called Model 1. The performance measures (NSE and RMSE) for each of the 464 rain gauges in the calibration and validation periods for Model 1 are plotted in Figs.5(a and b), respectively.
For comparison purposes, the experiment was repeated consid-ering (1) the same set of LSAPVs for each of the climatological re-gions, but with different SVR parameters for each region (Model 2), and (2) the same set of LSAPVs and the same SVR model for each region (Model 3). The performance measures in the calibration and validation periods for Models 2 and 3 are provided as boxplots in Figs.5(a and b)along with those for Model 1. The RMSE was the lowest and the NSE was closest to unity when different LSAPVs were considered and different SVR models were developed for each
Fig. 3. (Continued)
of the climatological regions. The performance decreased slightly (RMSE values increased and NSE values decreased) when the same set of LSAPVs was considered for all climatological regions but the SVR models were different for each region. The performance was inferior when the same set of LSAPVs was considered and the same SVR model was used for the entire study area. Better performance in the case of Model 1 can be attributed to the identification of clima-tological regions and the selection of appropriate attributes for each region. In addition, different SVR models were used to obtain future projections of rainfall for gauges located in each of those regions
because the rainfall generation process for each climatological re-gion was different. The performance decreased slightly in case of Model 2, in which different SVR models were selected but the LSAPVs were considered to be the same for the entire study area. The third approach was expected to provide inferior performance because the approach assumed that the same rainfall generation mechanism governs the rainfall for the entire study area.
Performance of the downscaled model (Model 1) in obtaining rainfall was compared with the high-resolution Copernicus Climate Change Service E-OBS v 19.0e gridded rainfall data obtained
6°W 6°W 8°W 8°W 10°W 10°W 56°N 56°N 54°N 54°N 52°N 52°N Cluster 1 Cluster 2 Cluster 3 Cluster 4 Cluster 5 Cluster 6 Republic of Ireland Great Britain 0 50 100KM
Fig. 4. Locations of climatological regions in Ireland.
Table 2. List of LSAPVs considered for rain gauges corresponding to each of the clusters
No. Variable
Region 1 Region 2 Region 3 Region 4 Region 5 Region 6
−ve +ve −ve +ve −ve +ve −ve +ve −ve +ve −ve +ve
1 q 150 850 200 100 200 100 200 100 200 100 200 100 2 r 200 700 250 500 250 600 250 600 250 925 250 600 3 t 700 200 600 — 700 200 500 200 700 — 700 200 4 u — 200 — 150 — 150 — 100 — 150 — 150 5 v — 500 — 925 — 1,000 — 925 — 1,000 — 850 6 w 400 — 500 — 400 — 400 — 700 — 400 — 7 z 1,000 — 925 — 1,000 — 1,000 — 1,000 — 1,000 — 8 tcw — — 1,000 — — — — — — — — — 9 msl 1,000 — 1,000 — 1,000 — 1,000 — 1,000 — 1,000 — 10 ishf — — — — — — — — — — — —
Note:–ve (+ve) denote rainfall negatively (positively) correlated to the LSAPV.
at 0.1° resolution. One representative station from each climatologi-cal region was selected, and the performance of the model predic-tions and the E-OBS gridded data was compared with the observed station rainfall data at the monthly scale. The performance of each station was plotted as a Taylor diagram (Taylor 2001) (Fig. S4). In the figure, the radial distance indicates the standard deviation of the monthly rainfall data, the azimuthal position provides the correlation coefficient value between the observed rainfall and the model-predicted/gridded rainfall, and the dotted line (shown in green) originating from the reference point (REF) provides the RMS distance (RMSD). In situations in which the predicted rainfall is closer to the observed rainfall, the standard deviation of the predicted rainfall is expected to be closer to that of the ob-served rainfall; the correlation should be closer to unity and the RMSD should be closer to zero. The figures indicate that the model-predicted rainfall had less error than that obtained from the gridded rainfall, except for the representative station located in Climatological region 5, for which the gridded rainfall was slightly better than the modeled value.
Once the SVR-based downscaling model is validated, it can be used to obtain future projections of mean monthly rainfall for all the
sites in the study area. Because the performance of Model 1 was superior compared with that of the other two models, future projec-tions of mean monthly rainfall were obtained based on this model only. For this purpose, simulations of National Aeronautics and Space Administration Goddard Institute for Space Studies General Circulation Model (NASA GISS GCM) coupled atmospheric– oceanic model data were used, corresponding to four Representative Concentration Pathways (RCPs). The resolution of the data was at a 2° × 2.5° grid scale, and they were regridded to a 2° × 2° grid scale that matched the ERA Interim reanalysis data spatial scale. The historical simulations of the GCM data from January 1979 to December 2005 were considered to perform a bias correction for the future projections of selected LSAPVs for the period January 2021–December 2050. The bias-corrected future projections of LSAPVs corresponding to each of the four RCP scenarios (RCP2.6, RCP4.5, RCP6.0, and RCP8.5) were used in the developed SVR-based downscaling model (Model 1) to obtain future projections of mean monthly rainfall at each of the 464 rain gauge sites. The change in the mean for three decades (2021–2030, 2031–2040, and 2041–2050) was estimated for each site in every cluster (clima-tological region) corresponding to each of the four RCP scenarios and were compared with the historical mean monthly rainfall. The results corresponding to each of the six climatological regions (clusters) are shown as boxplots in Figs.6(a–f). For Clusters 1, 4, and 6 (located in the eastern and southeastern part of the Republic of Ireland), the mean monthly rainfall is expected to increase in the future according to the RCP2.6 and RCP8.5 scenarios, whereas based on scenarios RCP 4.5 and RCP 6.0, the mean monthly rainfall is expected to increase in the next two decades (2021–2040) and to decrease after 2040. For Cluster 3, which is located in the central part of the country, the mean rainfall is expected to increase in the future according to the RCP8.5 scenario. For other locations and corresponding to all scenarios, the mean monthly rainfall is
Table 3. Optimal values of SVR parameters for each region using Model 1 and Model 2 Region Model 1 Model 2 γ C γ C 1 80 940 40 660 2 80 1,000 60 980 3 50 940 60 960 4 40 980 70 1,000 5 60 960 50 880 6 50 880 60 1,000
Fig. 5. Two performance measures obtained in calibration and validation by (1) identifying a set of attributes in each cluster and developing different SVR models for each cluster (Model 1); (2) considering the same set of attributes but developing different SVR models for each cluster (Model 2); (3) considering the same set of attributes and the same model for the entire study area (Model 3); (4) RVM-based downscaling considering different LSAPVs; and (5) LSTM-based downscaling considering different LSAPVs: (a) NSE; and (b) RMSE.
expected to decrease after 2040. RCP2.6 assumes that the global annual greenhouse gas (GHG) emissions (measured in equivalent CO2) will peak between 2010–2020 and decrease after 2020; in the case of the RCP4.5 scenario, the emissions peak around 2040 and then decrease; for RCP6.0, the emissions are assumed to peak around 2080 and then decline; whereas RCP8.5 assumes a continuous increase in emissions that continue to rise through-out the 21st century (Meinshausen et al. 2011). Chandler and Wheater (2002) considered generalized linear models to relate the changes in rainfall pattern at stations located in western Ireland with the North Atlantic Oscillation (NAO) index. Nolan et al. (2017) used a RCM model to investigate the effect of cli-mate change in the rainfall pattern across Ireland after the 2040s. They noted that the mean annual rainfall as well as rainfall in the summer and spring seasons are expected to decrease, whereas the frequency of heavy rainfall events in the winter months are sup-posed to increase.
Currently, with advancement of computational facilities, robust neural network–based techniques are gaining popularity for devel-oping complex relationships between high-dimensional predictor and predictand data sets. The relevance vector machine (RVM)-based downscaling approach has been considered in a few studies (Ghosh and Mujumdar 2008;Okkan and Inan 2015;Joshi et al. 2015;Deo et al. 2016) to downscale different hydroclimatological variables. The statistical framework of RVM is the same as that of the support vector machine algorithm; however, RVM considers an alternative functional algorithm that develops a probabilistic regres-sion between the predictors and predictant. Details of the RVM approach were given by Tipping (2001). Recently, deep learning– based techniques such as long short-term memory (LSTM) are becoming popular due to their better performance when trained with large amounts of data compared with other neural network– based approaches. This study further investigated the use of the LSTM algorithm in downscaling the rainfall data for the study area.
(a) (b) (c) (d) (e) (f) 2 2.5 3 3.5 2021-30 2031-40 2041-50 2021-30 2031-40 2041-50 2021-30 2031-40 2041-50 2021-30 2031-40 2041-50 OBS RCP2.6 RCP4.5 RCP6.0 RCP8.5
Mean Monthly Rainfall
3 4 5 6 7 8 9 2021-30 2031-40 2041-50 2021-30 2031-40 2041-50 2021-30 2031-40 2041-50 2021-30 2031-40 2041-50 OBS RCP2.6 RCP4.5 RCP6.0 RCP8.5
Mean Monthly Rainfall
2 2.5 3 3.5 4 4.5 2021-30 2031-40 2041-50 2021-30 2031-40 2041-50 2021-30 2031-40 2041-50 2021-30 2031-40 2041-50 OBS RCP2.6 RCP4.5 RCP6.0 RCP8.5
Mean Monthly Rainfall
2.5 3 3.5 4 4.5 2021-30 2031-40 2041-50 2021-30 2031-40 2041-50 2021-30 2031-40 2041-50 2021-30 2031-40 2041-50 OBS RCP2.6 RCP4.5 RCP6.0 RCP8.5
Mean Monthly Rainfall
2.5 3 3.5 4 4.5 5 5.5 6 6.5 2021-30 2031-40 2041-50 2021-30 2031-40 2041-50 2021-30 2031-40 2041-50 2021-30 2031-40 2041-50 OBS RCP2.6 RCP4.5 RCP6.0 RCP8.5
Mean Monthly Rainfall
2.5 3 3.5 4 4.5 2021-30 2031-40 2041-50 2021-30 2031-40 2041-50 2021-30 2031-40 2041-50 2021-30 2031-40 2041-50 OBS RCP2.6 RCP4.5 RCP6.0 RCP8.5
Mean Monthly Rainfall
Fig. 6. Mean projected monthly rainfall for each cluster averaged over 1 decade, corresponding to four climate change scenarios: (a) Cluster 1; (b) Cluster 2; (c) Cluster 3; (d) Cluster 4; (e) Cluster 5; and (f) Cluster 6. OBS = mean monthly rainfall of historically observed rainfall data.
The LSTM algorithm was considered to have 100 hidden units, and the maximum iteration was set to 250. Technical details of LSTM algorithm were given by Hochreiter and Schmidhuber (1997).
The performance of the RVM- and LSTM-based downscaling approaches was compared with that of the SVR-based downscaling approach for the Model 1 case, in which different sets of LSAPVs were selected for each climatological region. Only the Model 1 case was considered because the other two models (Model 2 and Model 3) consider unrealistic assumptions, leading to poor perfor-mance in downscaling rainfall. The perforperfor-mance of the two new approaches is provided in terms of two performance measures (RMSE and NSE) in Figs.5(a and b). Results indicate that the per-formance of LSTM downscaling was superior in the calibration period, followed by that of the RVM-based downscaling approach. However, the model performance of RVM and LSTM for the val-idation data set was found to be considerably inferior to that ob-tained using the SVR-based downscaling approach. Because the RVM considers the posterior distribution and LSTM can retrain the data set based on the model performance, prediction of rainfall for the calibration set using both approaches was better than that using the SVR-based approach. On the other hand, both RVM and LSTM require larger data sets for training the model, and because the monthly rainfall data for each station were limited, the perfor-mance of those models in the validation set was poor compared with that of the SVR-based model.
Conclusions
The paper presented a new approach to form climatological regions using a global K-means clustering algorithm where the rainfall generation mechanism can be considered to be similar. Sub-sequently, support vector regression–based statistical downscaling models were developed to obtain future projections of mean monthly rainfall at rain gauges located in each of those climatologi-cal regions. The effectiveness of the new approach was demon-strated through a case study of 464 rain gauges in the Republic of Ireland. The results indicated that the new approach provides better rainfall projections than the existing downscaling approaches which assume the same rainfall generation process for the entire study area. The newly developed downscaling model subsequently was used to obtain future projections of rainfall for the period 2021–2050 for four Representative Concentration Pathways sce-narios. In general, the mean rainfall is expected to increase until 2040 and then to decrease for all scenarios in the western part of the country, whereas in the eastern part of the country the rainfall is expected to increase after 2040 according to two climate change scenarios. Detailed analysis is underway to attribute the changes in the rainfall pattern in the Republic of Ireland. The performance of the rainfall downscaling technique proposed in the study depends on the developed climatological regions as well as on the model used to develop regression relationships between LSAPVs and his-torical rainfall. Research is underway to evaluate effectiveness of the climatological regions and to explore other nested downscaling approaches that can account for the physical rainfall generation phenomena.
Appendix. Details of Global K -Means Algorithm
The method attempts to minimize the following objective function:
FðZ; V∶KÞ ¼X N s¼1 XK i¼1 Iðzs∈ CiÞd2ðzs; viÞ ð19Þ
where Ci¼ ith hard cluster; I ¼ 1 if zs∈ Ci is true, and I ¼ 0 otherwise; V¼ ½v1; : : : ; vKT is a matrix containing centroids of K clusters, such that vi¼ ½vi;1; : : : ; vi;l ∈ ℜl; d2ð·; ·Þ = square of distance measure, which was considered to be Euclidean in this study; and vi;j= mean value of attribute j for cluster i.
1. Define Kmin and Kmax as the lower and the upper bounds, re-spectively, of the possible number of clusters. Initialize K to Kmin (¼ 2).
2. Compute the mean of N feature vectors fzs; s ¼ 1; : : : ; Ng [Eq. (20)] and choose it as the centroid of the first cluster. Set epoch to 1.
vi;j¼X N
s¼1
zs;j=N j ¼ 1; : : : ; l ð20Þ 3. Choose zepochas the centroid of the Kth cluster (i.e., vK¼ zepoch)
and set iteration count to 1.
4. Determine the Euclidean distance of each feature vector zsfrom the centroids of each K cluster and assign the vector to the clus-ter whose centroid is nearest to it.
5. Update the centroid of each cluster by computing the average of the feature vectors assigned to it. Then compute the value of the objective function Fð·Þ for the current iteration. If iteration count = 1, increment the count by 1 and proceed to Step 4. Otherwise, compute the difference in the value of Fð·Þ between the current and previous iterations. If the difference is suffi-ciently small (<0.001), store V and Fð·Þ and proceed to Step 6. Otherwise, increment iteration count by 1. If the count is less than or equal to a prespecified upper bound on the number of iterations (e.g., 500), proceed to Step 4; otherwise store and pro-ceed to Step 6.
6. Increment epoch by 1. If the resulting number is less than or equal to N, repeat Steps 3–5. Otherwise, proceed to Step 7. 7. Identify the epoch h that yielded the minimum value for the
objective function Fð·Þ and note the corresponding V as VoðKÞ ¼ ½vo
1; : : : ; voKTthat denotes the optimal cluster centroid matrix for the case in which the number of clusters is equal to K. 8. If K < Kmax, increment K by 1 and consider the first K − 1 clus-ter centroids as VoðK − 1Þ, set epoch to 1, and repeat Steps 3–7. Otherwise, proceed to Step 9.
9. From among the partitions resulting for all the K values ð2 ≤ K ≤ KmaxÞ, identify the optimal partition Kopt as that for which the value of the Davies–Bouldin cluster validity index (Davies and Bouldin 1979) is minimum. The index is a function of the ratio of the sum of within-cluster scatter to between-cluster separation. It is computed as
DBIðKÞ ¼ 1 K XK i¼1 max i; i ≠ r Scatteriþ Scatterr di;r ð21Þ where Scatteri and Scatterr = within-cluster scatter for ith and rth clusters, respectively. In general, Scatteri is computed as Scatteri¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPz
s∈Cikzs− v o ik2=Ni q
, di;r is the Euclidean dis-tance between the centroids of the ith and rth clusters, and Ni is the number of sites in cluster i.
Acknowledgments
The authors gratefully acknowledge the support of the Risk Analysis of Infrastructure Networks (RAIN) Project, Grant No. 608166, which is funded by European Union’s Seventh Framework Programme for Research, Technological Development and Demon-stration Activities.
Supplemental Data
Figs. S1–S4 are available online in the ASCE Library (www .ascelibrary.org).
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